Geometric QCD I: the Hodge-dual surface and quark confinement

This is the first of two papers presenting a geometric framework for Planar QCD ( N c → ∞). In this part, we establish the kinematic foundation of the theory by constructing the unique stable vacuum of the loop equation. We demonstrate that the Makeenko-Migdal loop equation admits a solution of the form W [ C ] = W p e r t [ C ] exp − κ S [ C ] , provided S [ C ] is a specific minimal surface possessing a self-dual area derivative. We prove that such a surface exists and corresponds to the Hodge-dual projection of a minimal surface in R 3 ⊗ R 4 . Crucially, this confinement mechanism relies on the self-duality of the area derivative—a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in D = 4 , distinguishing the resulting theory from standard string models which require higher critical dimensions. We relate the string tension parameter κ to the gluon condensate via the Operator Product Expansion. The dynamical quantization of the Fermi string on this rigid surface and the resulting meson spectrum are derived in the companion paper [1] .